copyright © 2009 pearson addison-wesley 2.2-1 2 acute angles and right triangle

Copyright © 2009 Pearson Addison-Wesley 2.2-1

2Acute Angles and Right Triangle

Copyright © 2009 Pearson Addison-Wesley

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2.2-2

Trigonometric Functions of Non-Acute Angles

2.2

Reference Angles ▪ Special Angles as Reference Angles ▪ Finding Angle Measures with Special Angles

Copyright © 2009 Pearson Addison-Wesley 2.2-3

Reference Angles

A reference angle for an angle θ is the positive acute angle made by the terminal side of angle θ and the x-axis.


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Caution

A common error is to find the reference angle by using the terminal side of θ and the y-axis.

The reference angle is always found with reference to the x-axis.


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Find the reference angle for an angle of 218°.

Example 1(a) FINDING REFERENCE ANGLES

The positive acute angle made by the terminal side of the angle and the x-axis is 218° – 180° = 38°.

For θ = 218°, the reference angle θ′ = 38°.


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Find the reference angle for an angle of 1387°.

Example 1(b) FINDING REFERENCE ANGLES

First find a coterminal angle between 0° and 360°.

Divide 1387 by 360 to get a quotient of about 3.9. Begin by subtracting 360° three times. 1387° – 3(360°) = 307°.

The reference angle for 307° (and thus for 1387°) is 360° – 307° = 53°.


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Find the values of the six trigonometric functions for 210°.

Example 2 FINDING TRIGONOMETRIC FUNCTION VALUES OF A QUADRANT III ANGLE

The reference angle for a 210° angle is 210° – 180° = 30°.

Choose point P on the terminal side of the angle so the distance from the origin to P is 2.


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Example 2 FINDING TRIGONOMETRIC FUNCTION VALUES OF A QUADRANT III ANGLE (continued)


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Finding Trigonometric Function Values Using the “Special Triangles”

Step 1 If θ is not between 0° and 360°, find a coterminal angle to θ that is. Do this by adding or subtracting 360° as many times as needed.

Step 2 Find the reference angle θ′.

Step 3 Using one of your “special triangles” and θ′, determine x, y, and r and then find the six trigonometric function values for θ. Be careful to use the right signs!


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Find the exact value of cos (–240°).

Example 3(a) FINDING TRIGONOMETRIC FUNCTION VALUES USING REFERENCE ANGLES

Since an angle of –240° is coterminal with an angle of –240° + 360° = 120°, the reference angle is 180° – 120° = 60°.

So


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Find the exact value of tan 675°.

Example 3(b) FINDING TRIGONOMETRIC FUNCTION VALUES USING REFERENCE ANGLES

Subtract 360° to find a coterminal angle between 0° and 360°: 675° – 360° = 315°.

copyright © 2009 pearson addison-wesley 2.2-1 2 acute angles and right triangle

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